Internal problem ID [9005]
Internal file name [OUTPUT/7940_Monday_June_06_2022_12_58_39_AM_24144226/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 671.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {\left (y^{2} x +1\right )^{2}}{y x^{4}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } y x^{4}-y^{4} x^{2}-2 y^{2} x -1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{4} x^{2}+2 y^{2} x +1}{y x^{4}} \end {array} \]
Maple trace Kovacic algorithm successful
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -4*y(x)/x^6-2*(x^2-2)*(diff(y(x), x))/x^3, y(x)` *** Sublevel 2 *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Group is reducible or imprimitive <- Kovacics algorithm successful <- differential order: 1; linearization to 2nd order successful`
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 197
dsolve(diff(y(x),x) = (x*y(x)^2+1)^2/y(x)/x^4,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {2}\, {\mathrm e}^{\frac {\sqrt {2}\, x +1}{x^{2}}} \sqrt {-\left ({\mathrm e}^{\frac {2 \sqrt {2}}{x}}+c_{1} \right ) x \left (\left (2-\sqrt {2}\, x \right ) {\mathrm e}^{\frac {2 \sqrt {2}}{x}}+c_{1} \left (\sqrt {2}\, x +2\right )\right ) {\mathrm e}^{-\frac {2}{x^{2}}} {\mathrm e}^{-\frac {2 \sqrt {2}}{x}}}}{2 x \left ({\mathrm e}^{\frac {2 \sqrt {2}}{x}}+c_{1} \right )} \\ y \left (x \right ) &= \frac {\sqrt {2}\, {\mathrm e}^{\frac {\sqrt {2}\, x +1}{x^{2}}} \sqrt {-\left ({\mathrm e}^{\frac {2 \sqrt {2}}{x}}+c_{1} \right ) x \left (\left (2-\sqrt {2}\, x \right ) {\mathrm e}^{\frac {2 \sqrt {2}}{x}}+c_{1} \left (\sqrt {2}\, x +2\right )\right ) {\mathrm e}^{-\frac {2}{x^{2}}} {\mathrm e}^{-\frac {2 \sqrt {2}}{x}}}}{2 x \left ({\mathrm e}^{\frac {2 \sqrt {2}}{x}}+c_{1} \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 14.007 (sec). Leaf size: 206
DSolve[y'[x] == (1 + x*y[x]^2)^2/(x^4*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {\frac {-\sqrt {2} x+\left (\sqrt {2} x-2\right ) e^{\frac {2 \sqrt {2} (1+c_1 x)}{x}}-2}{x}}}{\sqrt {2} \sqrt {1+e^{\frac {2 \sqrt {2} (1+c_1 x)}{x}}}} \\ y(x)\to \frac {\sqrt {\frac {-\sqrt {2} x+\left (\sqrt {2} x-2\right ) e^{\frac {2 \sqrt {2} (1+c_1 x)}{x}}-2}{x}}}{\sqrt {2} \sqrt {1+e^{\frac {2 \sqrt {2} (1+c_1 x)}{x}}}} \\ y(x)\to -\sqrt {-\frac {1}{x}-\frac {1}{\sqrt {2}}} \\ y(x)\to \sqrt {-\frac {1}{x}-\frac {1}{\sqrt {2}}} \\ \end{align*}